Rode YZR500 machines since the 1994 British GP. A full-time participant in the GP500 from 1995, Abe achieved three GP victories: in 1996 in the Japan GP, in 1999 in Rio de Janeiro, and in 2000 in the Japanese event. From 2005 he raced in the World Superbike series, but from 2007 returned to racing in Japan. In October of the same year, however, he lost his life in a traffic accident. In 1993 at the All Japan Road Race Championship, won the 500cc title (Honda) and in 1994 entered the Japan GP, where he led Mick Doohan (Honda) and Kevin Schwantz (Suzuki) until finally taking a fall with three laps before the finish. That year Abe rode for Honda in the All Japan Superbike Championship, but during the season switched to a YZR500 in place of Daryl Beattie who was injured in the GP. Abe flew to the United Kingdom where he rode in three races. From the following year, 1995, he became a full-time rider.

In 2002, the first year of the MotoGP, he achieved the highest ranking of any 2-stroke rider wiht 6th. Out of 146 GP races, he pulled out of finals only twice, due to injuries in the qualifying rounds. These were his first race on a YZR500 (1994, United Kingdom) and his first race on a YZR-M1 (2002, Australia). In the other 144 races, Abe rode through to the finals. From 2005, rode in the World Superbike Championship, and in both that year and the next, achieved rankings of 13th. In 2007, returned to Japanese racing after a hiatus of 13 years. Entered the Suzuka 8 Hours Endurance Race for the first time and was placed 9th.